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fundamental product theorem in topological K-theory

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cohomology

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Idea

For XX a compact Hausdorff space The fundamental product theorem in topological K-theory identifies

  1. the topological K-theory-ring K(X×S 2)K(X \times S^2) of the product topological space X×S 2X \times S^2 with the 2-sphere S 2S^2 ;

  2. the K-theory ring K(X)K(X) of the original space XX with a generator HH for the basic line bundle on the 2-sphere adjoined:

K(X) [H]/(H1) 2K(X×S 2). K(X) \otimes_{\mathbb{Z}} \mathbb{Z}[H]/(H-1)^2 \overset{\simeq}{\longrightarrow} K(X \times S^2) \,.

This theorem in particular serves as a substantial step in a proof of Bott periodicity for topological K-theory (cor below).

The usual proof proceeds by

  1. realizing all vector bundles on X×S 2X \times S^2 via an XX-parameterized clutching construction;

  2. showing that all the clutching functions are homotopic to those that are Laurent polynomials as functions on S 1S^1, hence products of a polynomial clutching pp functions with a monomial z nz^{-n} of negative power;

  3. observing that the bundle corresponding to a clutching function of the form fz nf z^n is equivalent to the bundle corresponding to ff and tensored with the nnth tensor product of vector bundles-power of the basic complex line bundle on the 2-sphere;

  4. showing that some direct sum of vector bundles of the vector bundle corresponding to a polynomial clutching function with one coming from a trivial clutching function is given by a linear clutching function;

  5. showing that bundles coming from linear clutching functions are direct sums of one coming from a trivial clutching function with the one coming from the homogeneously linear part;

Applying these steps to a vector bundle on X×S 2X \times S^2 yields a virtual sum of external tensor products of vector bundles of bundles on XX with powers of the basic complex line bundle on the 2-sphere. This means that the function in the fundamental product theorem is surjective. By similar means one shows that it is also injective.

Statement

For S 2 3S^2 \subset \mathbb{R}^3 the 2-sphere with its Euclidean subspace topology, write hh for the basic line bundle on the 2-sphere. Its image in the topological K-theory ring K(S 2)K(S^2) satisfies the relation

2h=h 2+1(h1) 2=0 2 h = h^2 + 1 \;\;\Leftrightarrow\;\; (h-1)^2 = 0

(by this prop.).

Notice that h1h-1 is the image of hh in the reduced K-theory K˜(X)\tilde K(X) of S 2S^2 under the splitting K(X)K˜(X)K(X) \simeq \tilde K(X) \oplus \mathbb{Z} (by this prop.). This element

h1K˜ (S 2) h - 1 \in \tilde K_{\mathbb{C}}(S^2)

is called the Bott element of complex topological K-theory.

It follows that there is a ring homomorphism of the form

[h]/((h1) 2) K(S 2) h AAA h \array{ \mathbb{Z}[h]/\left( (h-1)^2 \right) &\overset{}{\longrightarrow}& K(S^2) \\ h &\overset{\phantom{AAA}}{\mapsto}& h }

from the polynomial ring in one abstract generator, quotiented by this relation, to the topological K-theory ring.

More generally, for XX a topological space, then this induces the composite ring homomorphism

Φ: K(X)[h]/((h1) 2) K(X)K(S 2) K(X×S 2) (E,h) AAA (E,H) AAA (π X *E)(π S 2 *H) \array{ \Phi \colon & K(X) \otimes \mathbb{Z}[h]/((h-1)^2) & \longrightarrow & K(X) \otimes K(S^2) & \overset{\boxtimes}{\longrightarrow} & K(X \times S^2) \\ & (E, h) &\overset{\phantom{AAA} }{\mapsto}& (E,H) &\overset{\phantom{AAA}}{\mapsto}& (\pi_{X}^\ast E) \cdot (\pi_{S^2}^\ast H) }

to the topological K-theory ring of the product topological space X×S 2X \times S^2, where the second map \boxtimes is the external tensor product of vector bundles.

Proposition

(fundamental product theorem in topological K-theory)

For XX a compact Hausdorff space, then ring homomorphism Φ:K(X)[h]/((h1) 2)K(X×S 2)\Phi \colon K(X) \otimes \mathbb{Z}[h]/((h-1)^2) \longrightarrow K(X \times S^2) is an isomorphism.

(e.g. Hatcher, theorem 2.2)

Remark

More generally, for LXL\to X a complex line bundle with class lK(X)l \in K(X) and with P(1L)P(1 \oplus L) denoting its projective bundle then

K(X)[h]/((h1)(lh1))K(P(1L)) K(X)[h]/((h-1)(l \cdot h -1)) \simeq K(P(1 \oplus L))

(e.g. Wirthmuller 12, p. 17)

As a special case this implies the first statement above:

For X=*X = \ast the product theorem prop. says in particular that the first of the two morphisms in the composite is an isomorphism (example below) and hence by the two-out-of-three-property for isomorphisms it follows that

Corollary

(external product theorem)

For XX a compact Hausdorff space we have that the external tensor product of vector bundles with vector bundles on the 2-sphere

:K(X)K(S 2)K(X×S 2) \boxtimes \;\colon\; K(X) \otimes K(S^2) \overset{\simeq}{\longrightarrow} K(X \times S^2)

is an isomorphism in topological K-theory.

Bott periodicity

When restricted to reduced K-theory then the external product theorem (cor. ) yields the statement of Bott periodicity of topological K-theory:

Corollary

(Bott periodicity)

Let XX be a pointed compact Hausdorff space.

Then there is an isomorphism of reduced K-theory

(h1)˜():K˜(X)K˜(Σ 2X) (h-1) \widetilde \boxtimes (-) \;\colon\; \tilde K(X) \overset{\simeq}{\longrightarrow} \tilde K(\Sigma^2 X)

from that of XX to that of its double suspension Σ 2X\Sigma^2 X.

Proof

By this example there is for any two pointed compact Hausdorff spaces XX and YY an isomorphism

K˜(Y×X)K˜(YX)K˜(Y)K˜(X) \tilde K(Y \times X) \simeq \tilde K(Y \wedge X) \oplus \tilde K(Y) \oplus \tilde K(X)

relating the reduced K-theory of the product topological space with that of the smash product.

Using this and the fact that for any pointed compact Hausdorff space ZZ we have K(Z)K˜(Z)K(Z) \simeq \tilde K(Z) \oplus \mathbb{Z} (this prop.) the isomorphism of the external product theorem (cor. )

K(S 2)K(X)K(S 2×X) K(S^2) \otimes K(X) \underoverset{\simeq}{\boxtimes}{\longrightarrow} K(S^2 \times X)

becomes

(K˜(S 2))(K˜(X))(K˜(S 2×X))(K˜(S 2X)K˜(S 2)K˜(X)). \left( \tilde K(S^2) \oplus \mathbb{Z} \right) \otimes \left( \tilde K(X) \oplus \mathbb{Z} \right) \;\simeq\; \left( \tilde K(S^2 \times X) \oplus \mathbb{Z} \right) \simeq \left( \tilde K(S^2 \wedge X) \oplus \tilde K(S^2) \oplus \tilde K(X) \oplus \mathbb{Z} \right) \,.

Multiplying out and chasing through the constructions to see that this reduces to an isomorphism on the common summand K˜(S 2)K˜(X)\tilde K(S^2) \oplus \tilde K(X) \oplus \mathbb{Z}, this yields an isomorphism of the form

K˜(S 2)K˜(X)˜K˜(S 2X)=K˜(Σ 2X), \tilde K(S^2) \otimes \tilde K(X) \underoverset{\simeq}{\widetilde \boxtimes}{\longrightarrow} \tilde K(S^2 \wedge X) = \tilde K(\Sigma^2 X) \,,

where on the right we used that smash product with the 2-sphere is the same as double suspension.

Finally there is an isomorphism

β K˜ (S 2) 1 AAA (h1) \array{ \mathbb{Z} &\underoverset{\simeq}{ \beta }{\longrightarrow}& \tilde K_{\mathbb{C}}(S^2) \\ 1 &\overset{\phantom{AAA}}{\mapsto}& (h-1) }

(example ). The composite

K˜ (X) K˜ (X)βidK˜ (S 2)K˜ (X)˜ K˜ (S 2X)=K˜ (Σ 2X) Erk x(E) AAAA (h1)˜(Erk x(E)) \array{ \tilde K_{\mathbb{C}}(X) & \simeq \mathbb{Z} \otimes \tilde K_{\mathbb{C}}(X) \overset{ \beta \otimes id }{\longrightarrow} \tilde K_{\mathbb{C}}(S^2) \otimes \tilde K_{\mathbb{C}}(X) \underoverset{\simeq}{\widetilde \boxtimes}{\longrightarrow} & \tilde K_{\mathbb{C}}(S^2 \wedge X) = \tilde K_{\mathbb{C}}(\Sigma^2 X) \\ E - rk_x(E) &\overset{\phantom{AAAA}}{\mapsto}& (h-1) \widetilde \boxtimes (E - rk_x(E)) }

is the isomorphism to be established.

Examples

Example

(topological K-theory ring of the 2-sphere)

For X=*X = \ast the point space, the fundamental product theorem states that the homomorphism

[h]/((h1) 2) K(S 2) h h \array{ \mathbb{Z}[h]/((h-1)^2) &\longrightarrow& K(S^2) \\ h &\mapsto& h }

is an isomorphism.

This means that the relation (h1) 2=0(h-1)^2 = 0 satisfied by the basic line bundle on the 2-sphere (this prop.) is the only relation is satisfies in topological K-theory.

Notice that the underlying abelian group of [h]/((h1) 2)\mathbb{Z}[h]/((h-1)^2) is two direct sum copies of the integers,

K(S 2)=1,h K(S^2) \simeq \mathbb{Z} \oplus \mathbb{Z} = \langle 1, h\rangle

one copy spanned by the trivial complex line bundle on the 2-sphere, the other spanned by the basic complex line bundle on the 2-sphere. (In contrast, the underlying abelian group of the polynomial ring [h]\mathbb{R}[h] has infinitely many copies of \mathbb{Z}, one for each h nh^n, for nn \in \mathbb{N}).

It follows (by this prop.) that the reduced K-theory group of the 2-sphere is

K˜(S 2). \tilde K(S^2) \simeq \mathbb{Z} \,.

References

Expositions include:

Last revised on June 21, 2017 at 06:09:07. See the history of this page for a list of all contributions to it.